Problem: Onur is floating freely in outer space with a propulsion thruster to help him counteract the pull of gravity. The gravitational forces pulling on his body are negligible except for those from the nearest planet (described by force vector ${\vec p}\,$ ) and the nearest star (described by force vector ${\vec s}\,$ ). Onur, the planet, and the star are in the same plane, so we can describe the gravitational forces as two-dimensional vectors. ${\vec{p}} = (4,6)$ ${\vec{s}} = (1,7)$ (Forces are given in newtons, $\text{N}$.) Assuming Onur wants the net gravitational force pulling on his body to be zero, with what force should he engage his propulsion thruster?
Explanation: Whenever forces are pulling in different directions, they tend to partially cancel each other out. We're looking for a force for Onur's propulsion thruster that will perfectly cancel the gravitational forces of ${\vec p}$ and ${\vec s}$. This canceling effect is exactly what happens when we add vectors. In order for ${\vec t}$ to perfectly cancel out the effects of ${\vec p}$ and ${\vec s}$, the following relationship must be true: ${\vec t} + {\vec p} + {\vec s} = (0,0)$. So if ${\vec p} + {\vec s}$ is $(4,6) + (1,7) = (5, 13)$, then ${\vec t}$ must be $(-5, -13)$. We can find the magnitude of ${\vec{t}} $ using the Pythagorean theorem, which will tell us the force at which Onur should engage the thruster. $\begin{aligned} \|{\vec{t}} \|^2 &= (-5)^2 + (-13)^2\\\\ \|{\vec{t}} \| &= \sqrt{25 + 169}\\\\ \|{\vec{t}} \| &= \sqrt{194}\\\\ \|{\vec{t}} \| &\approx 13.9 \text{ N} \end{aligned}$ Finding the direction of ${\vec t}$ will tell us in what direction Onur should engage his thruster. ${\vec t}$ is pointing in the third quadrant with an $x$ -component of $-5$ and a $y$ -component of $-13$. We can find the direction of any vector $\vec v$ in the third quadrant using the arctangent function and adding $\pi$. $\begin{aligned} \tan \theta &= \dfrac{ y}{ x}\\ \\ \tan \theta &= \dfrac{-13}{-5}\\\\ \theta &= \arctan{\left( \dfrac{13}{5} \right)} \\\\ \theta&\approx 1.2036\,\text{rad} \end{aligned}$ Adding $\pi$, we get $4.3\,\text{rad}$. Onur should engage his propulsion thruster to $13.9 \text{ N}$. Onur's propulsion thruster should exert force at an angle of $4.3$ radians.